3.171 \(\int (c+d x)^4 \cos (a+b x) \cot ^2(a+b x) \, dx\)

Optimal. Leaf size=299 \[ -\frac {24 i d^4 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^4 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^5}-\frac {24 d^4 \sin (a+b x)}{b^5}-\frac {24 d^3 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \cos (a+b x)}{b^4}+\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^3}+\frac {12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac {4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac {8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {(c+d x)^4 \csc (a+b x)}{b} \]

[Out]

-8*d*(d*x+c)^3*arctanh(exp(I*(b*x+a)))/b^2+24*d^3*(d*x+c)*cos(b*x+a)/b^4-4*d*(d*x+c)^3*cos(b*x+a)/b^2-(d*x+c)^
4*csc(b*x+a)/b+12*I*d^2*(d*x+c)^2*polylog(2,-exp(I*(b*x+a)))/b^3-12*I*d^2*(d*x+c)^2*polylog(2,exp(I*(b*x+a)))/
b^3-24*d^3*(d*x+c)*polylog(3,-exp(I*(b*x+a)))/b^4+24*d^3*(d*x+c)*polylog(3,exp(I*(b*x+a)))/b^4-24*I*d^4*polylo
g(4,-exp(I*(b*x+a)))/b^5+24*I*d^4*polylog(4,exp(I*(b*x+a)))/b^5-24*d^4*sin(b*x+a)/b^5+12*d^2*(d*x+c)^2*sin(b*x
+a)/b^3-(d*x+c)^4*sin(b*x+a)/b

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Rubi [A]  time = 0.29, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {4408, 3296, 2637, 4410, 4183, 2531, 6609, 2282, 6589} \[ -\frac {24 d^3 (c+d x) \text {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \text {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}+\frac {12 i d^2 (c+d x)^2 \text {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \text {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac {24 i d^4 \text {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^4 \text {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^5}+\frac {12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}+\frac {24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac {8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac {24 d^4 \sin (a+b x)}{b^5}-\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {(c+d x)^4 \csc (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^4*Cos[a + b*x]*Cot[a + b*x]^2,x]

[Out]

(-8*d*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b^2 + (24*d^3*(c + d*x)*Cos[a + b*x])/b^4 - (4*d*(c + d*x)^3*Cos[a
+ b*x])/b^2 - ((c + d*x)^4*Csc[a + b*x])/b + ((12*I)*d^2*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 - ((12
*I)*d^2*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (24*d^3*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^4 + (
24*d^3*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^4 - ((24*I)*d^4*PolyLog[4, -E^(I*(a + b*x))])/b^5 + ((24*I)*d^
4*PolyLog[4, E^(I*(a + b*x))])/b^5 - (24*d^4*Sin[a + b*x])/b^5 + (12*d^2*(c + d*x)^2*Sin[a + b*x])/b^3 - ((c +
d*x)^4*Sin[a + b*x])/b

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2531

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4183

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*ArcTanh[E^(I*(e + f*
x))])/f, x] + (-Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[(d*m)/f, Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4408

Int[Cos[(a_.) + (b_.)*(x_)]^(n_.)*Cot[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[
(c + d*x)^m*Cos[a + b*x]^n*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Cos[a + b*x]^(n - 2)*Cot[a + b*x]^p, x]
/; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 4410

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp
[((c + d*x)^m*Csc[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Fr
eeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6609

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[((e + f*x)^m*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p])/(b*c*p*Log[F]), x] - Dist[(f*m)/(b*c*p*Log[F]), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rubi steps

\begin {align*} \int (c+d x)^4 \cos (a+b x) \cot ^2(a+b x) \, dx &=-\int (c+d x)^4 \cos (a+b x) \, dx+\int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx\\ &=-\frac {(c+d x)^4 \csc (a+b x)}{b}-\frac {(c+d x)^4 \sin (a+b x)}{b}+\frac {(4 d) \int (c+d x)^3 \csc (a+b x) \, dx}{b}+\frac {(4 d) \int (c+d x)^3 \sin (a+b x) \, dx}{b}\\ &=-\frac {8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac {4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac {(c+d x)^4 \csc (a+b x)}{b}-\frac {(c+d x)^4 \sin (a+b x)}{b}+\frac {\left (12 d^2\right ) \int (c+d x)^2 \cos (a+b x) \, dx}{b^2}-\frac {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (12 d^2\right ) \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac {8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac {4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^3}+\frac {12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {\left (24 i d^3\right ) \int (c+d x) \text {Li}_2\left (-e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (24 i d^3\right ) \int (c+d x) \text {Li}_2\left (e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (24 d^3\right ) \int (c+d x) \sin (a+b x) \, dx}{b^3}\\ &=-\frac {8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}+\frac {24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^3}-\frac {24 d^3 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^4}+\frac {12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {\left (24 d^4\right ) \int \cos (a+b x) \, dx}{b^4}+\frac {\left (24 d^4\right ) \int \text {Li}_3\left (-e^{i (a+b x)}\right ) \, dx}{b^4}-\frac {\left (24 d^4\right ) \int \text {Li}_3\left (e^{i (a+b x)}\right ) \, dx}{b^4}\\ &=-\frac {8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}+\frac {24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^3}-\frac {24 d^3 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^4}-\frac {24 d^4 \sin (a+b x)}{b^5}+\frac {12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac {(c+d x)^4 \sin (a+b x)}{b}-\frac {\left (24 i d^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}+\frac {\left (24 i d^4\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}\\ &=-\frac {8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}+\frac {24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac {4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac {(c+d x)^4 \csc (a+b x)}{b}+\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac {12 i d^2 (c+d x)^2 \text {Li}_2\left (e^{i (a+b x)}\right )}{b^3}-\frac {24 d^3 (c+d x) \text {Li}_3\left (-e^{i (a+b x)}\right )}{b^4}+\frac {24 d^3 (c+d x) \text {Li}_3\left (e^{i (a+b x)}\right )}{b^4}-\frac {24 i d^4 \text {Li}_4\left (-e^{i (a+b x)}\right )}{b^5}+\frac {24 i d^4 \text {Li}_4\left (e^{i (a+b x)}\right )}{b^5}-\frac {24 d^4 \sin (a+b x)}{b^5}+\frac {12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac {(c+d x)^4 \sin (a+b x)}{b}\\ \end {align*}

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Mathematica [B]  time = 1.71, size = 798, normalized size = 2.67 \[ \frac {\csc (a+b x) \left (-3 c^4 b^4-3 d^4 x^4 b^4-12 c d^3 x^3 b^4-18 c^2 d^2 x^2 b^4-12 c^3 d x b^4+c^4 \cos (2 (a+b x)) b^4+d^4 x^4 \cos (2 (a+b x)) b^4+4 c d^3 x^3 \cos (2 (a+b x)) b^4+6 c^2 d^2 x^2 \cos (2 (a+b x)) b^4+4 c^3 d x \cos (2 (a+b x)) b^4-16 d^4 x^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-48 c d^3 x^2 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-16 c^3 d \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-48 c^2 d^2 x \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-4 d^4 x^3 \sin (2 (a+b x)) b^3-12 c d^3 x^2 \sin (2 (a+b x)) b^3-4 c^3 d \sin (2 (a+b x)) b^3-12 c^2 d^2 x \sin (2 (a+b x)) b^3+12 c^2 d^2 b^2+12 d^4 x^2 b^2+24 c d^3 x b^2-12 c^2 d^2 \cos (2 (a+b x)) b^2-12 d^4 x^2 \cos (2 (a+b x)) b^2-24 c d^3 x \cos (2 (a+b x)) b^2+24 i d^2 (c+d x)^2 \text {Li}_2(-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b^2-24 i d^2 (c+d x)^2 \text {Li}_2(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^2-48 c d^3 \text {Li}_3(-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b-48 d^4 x \text {Li}_3(-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b+48 c d^3 \text {Li}_3(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b+48 d^4 x \text {Li}_3(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b+24 c d^3 \sin (2 (a+b x)) b+24 d^4 x \sin (2 (a+b x)) b-24 d^4+24 d^4 \cos (2 (a+b x))-48 i d^4 \text {Li}_4(-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x)+48 i d^4 \text {Li}_4(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x)\right )}{2 b^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(c + d*x)^4*Cos[a + b*x]*Cot[a + b*x]^2,x]

[Out]

(Csc[a + b*x]*(-3*b^4*c^4 + 12*b^2*c^2*d^2 - 24*d^4 - 12*b^4*c^3*d*x + 24*b^2*c*d^3*x - 18*b^4*c^2*d^2*x^2 + 1
2*b^2*d^4*x^2 - 12*b^4*c*d^3*x^3 - 3*b^4*d^4*x^4 + b^4*c^4*Cos[2*(a + b*x)] - 12*b^2*c^2*d^2*Cos[2*(a + b*x)]
+ 24*d^4*Cos[2*(a + b*x)] + 4*b^4*c^3*d*x*Cos[2*(a + b*x)] - 24*b^2*c*d^3*x*Cos[2*(a + b*x)] + 6*b^4*c^2*d^2*x
^2*Cos[2*(a + b*x)] - 12*b^2*d^4*x^2*Cos[2*(a + b*x)] + 4*b^4*c*d^3*x^3*Cos[2*(a + b*x)] + b^4*d^4*x^4*Cos[2*(
a + b*x)] - 16*b^3*c^3*d*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 48*b^3*c^2*d^2*x*ArcTanh[Cos[a
+ b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 48*b^3*c*d^3*x^2*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] -
16*b^3*d^4*x^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] + (24*I)*b^2*d^2*(c + d*x)^2*PolyLog[2, -C
os[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] - (24*I)*b^2*d^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*
x]]*Sin[a + b*x] - 48*b*c*d^3*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] - 48*b*d^4*x*PolyLog[3,
-Cos[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] + 48*b*c*d^3*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*
x] + 48*b*d^4*x*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - (48*I)*d^4*PolyLog[4, -Cos[a + b*x] -
I*Sin[a + b*x]]*Sin[a + b*x] + (48*I)*d^4*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 4*b^3*c^3*
d*Sin[2*(a + b*x)] + 24*b*c*d^3*Sin[2*(a + b*x)] - 12*b^3*c^2*d^2*x*Sin[2*(a + b*x)] + 24*b*d^4*x*Sin[2*(a + b
*x)] - 12*b^3*c*d^3*x^2*Sin[2*(a + b*x)] - 4*b^3*d^4*x^3*Sin[2*(a + b*x)]))/(2*b^5)

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fricas [C]  time = 0.63, size = 1233, normalized size = 4.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cos(b*x+a)*cot(b*x+a)^2,x, algorithm="fricas")

[Out]

-(2*b^4*d^4*x^4 + 8*b^4*c*d^3*x^3 + 2*b^4*c^4 - 12*b^2*c^2*d^2 - 12*I*d^4*polylog(4, cos(b*x + a) + I*sin(b*x
+ a))*sin(b*x + a) + 12*I*d^4*polylog(4, cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) - 12*I*d^4*polylog(4, -co
s(b*x + a) + I*sin(b*x + a))*sin(b*x + a) + 12*I*d^4*polylog(4, -cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) +
24*d^4 + 12*(b^4*c^2*d^2 - b^2*d^4)*x^2 - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + b^4*c^4 - 12*b^2*c^2*d^2 + 24*d^4
+ 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 4*(b^4*c^3*d - 6*b^2*c*d^3)*x)*cos(b*x + a)^2 + 4*(b^3*d^4*x^3 + 3*b^3*c*d
^3*x^2 + b^3*c^3*d - 6*b*c*d^3 + 3*(b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)*sin(b*x + a) - (-6*I*b^2*d^4*x^2 -
12*I*b^2*c*d^3*x - 6*I*b^2*c^2*d^2)*dilog(cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - (6*I*b^2*d^4*x^2 + 12*
I*b^2*c*d^3*x + 6*I*b^2*c^2*d^2)*dilog(cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) - (-6*I*b^2*d^4*x^2 - 12*I*
b^2*c*d^3*x - 6*I*b^2*c^2*d^2)*dilog(-cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - (6*I*b^2*d^4*x^2 + 12*I*b^
2*c*d^3*x + 6*I*b^2*c^2*d^2)*dilog(-cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 2*(b^3*d^4*x^3 + 3*b^3*c*d^3
*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d)*log(cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a) + 2*(b^3*d^4*x^3 + 3*b
^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d)*log(cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a) - 2*(b^3*c^3*d
- 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/2)*sin(b*x + a) -
2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*log(-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2)*s
in(b*x + a) - 2*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + 3*a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + a^3*d^4)*
log(-cos(b*x + a) + I*sin(b*x + a) + 1)*sin(b*x + a) - 2*(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + 3*
a*b^2*c^2*d^2 - 3*a^2*b*c*d^3 + a^3*d^4)*log(-cos(b*x + a) - I*sin(b*x + a) + 1)*sin(b*x + a) - 12*(b*d^4*x +
b*c*d^3)*polylog(3, cos(b*x + a) + I*sin(b*x + a))*sin(b*x + a) - 12*(b*d^4*x + b*c*d^3)*polylog(3, cos(b*x +
a) - I*sin(b*x + a))*sin(b*x + a) + 12*(b*d^4*x + b*c*d^3)*polylog(3, -cos(b*x + a) + I*sin(b*x + a))*sin(b*x
+ a) + 12*(b*d^4*x + b*c*d^3)*polylog(3, -cos(b*x + a) - I*sin(b*x + a))*sin(b*x + a) + 8*(b^4*c^3*d - 3*b^2*c
*d^3)*x)/(b^5*sin(b*x + a))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{4} \cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cos(b*x+a)*cot(b*x+a)^2,x, algorithm="giac")

[Out]

integrate((d*x + c)^4*cos(b*x + a)*cot(b*x + a)^2, x)

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maple [B]  time = 0.16, size = 1056, normalized size = 3.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^4*cos(b*x+a)*cot(b*x+a)^2,x)

[Out]

8/b^5*d^4*a^3*arctanh(exp(I*(b*x+a)))-24/b^4*d^3*c*polylog(3,-exp(I*(b*x+a)))+24/b^4*d^3*c*polylog(3,exp(I*(b*
x+a)))-8/b^2*d*c^3*arctanh(exp(I*(b*x+a)))-24/b^4*d^4*polylog(3,-exp(I*(b*x+a)))*x+24/b^4*d^4*polylog(3,exp(I*
(b*x+a)))*x-24*I*d^4*polylog(4,-exp(I*(b*x+a)))/b^5+24*I/b^3*d^3*c*polylog(2,-exp(I*(b*x+a)))*x-24*I/b^3*d^3*c
*polylog(2,exp(I*(b*x+a)))*x+24*I*d^4*polylog(4,exp(I*(b*x+a)))/b^5+1/2*I*(d^4*x^4*b^4+4*b^4*c*d^3*x^3+6*b^4*c
^2*d^2*x^2+4*b^4*c^3*d*x+4*I*b^3*d^4*x^3+b^4*c^4-12*b^2*d^4*x^2+12*I*b^3*c*d^3*x^2-24*b^2*c*d^3*x+12*I*b^3*c^2
*d^2*x-12*c^2*d^2*b^2+4*I*b^3*c^3*d-24*I*b*d^4*x+24*d^4-24*I*b*c*d^3)/b^5*exp(I*(b*x+a))-2*I*(d^4*x^4+4*c*d^3*
x^3+6*c^2*d^2*x^2+4*c^3*d*x+c^4)*exp(I*(b*x+a))/b/(exp(2*I*(b*x+a))-1)-1/2*I*(d^4*x^4*b^4+4*b^4*c*d^3*x^3+6*b^
4*c^2*d^2*x^2+4*b^4*c^3*d*x-4*I*b^3*d^4*x^3+b^4*c^4-12*b^2*d^4*x^2-12*I*b^3*c*d^3*x^2-24*b^2*c*d^3*x-12*I*b^3*
c^2*d^2*x-12*c^2*d^2*b^2-4*I*b^3*c^3*d+24*I*b*d^4*x+24*d^4+24*I*b*c*d^3)/b^5*exp(-I*(b*x+a))+12/b^2*d^2*c^2*ln
(1-exp(I*(b*x+a)))*x+12/b^3*d^2*c^2*ln(1-exp(I*(b*x+a)))*a-12/b^2*d^2*c^2*ln(exp(I*(b*x+a))+1)*x-12/b^3*d^2*c^
2*ln(exp(I*(b*x+a))+1)*a+12/b^2*d^3*c*ln(1-exp(I*(b*x+a)))*x^2-12/b^4*d^3*c*ln(1-exp(I*(b*x+a)))*a^2-12/b^2*d^
3*c*ln(exp(I*(b*x+a))+1)*x^2+12/b^4*d^3*c*ln(exp(I*(b*x+a))+1)*a^2+4/b^2*d^4*ln(1-exp(I*(b*x+a)))*x^3+4/b^5*d^
4*ln(1-exp(I*(b*x+a)))*a^3-4/b^2*d^4*ln(exp(I*(b*x+a))+1)*x^3-4/b^5*d^4*ln(exp(I*(b*x+a))+1)*a^3-24/b^4*d^3*c*
a^2*arctanh(exp(I*(b*x+a)))+24/b^3*d^2*c^2*a*arctanh(exp(I*(b*x+a)))+12*I/b^3*d^4*polylog(2,-exp(I*(b*x+a)))*x
^2-12*I/b^3*d^4*polylog(2,exp(I*(b*x+a)))*x^2+12*I/b^3*d^2*c^2*polylog(2,-exp(I*(b*x+a)))-12*I/b^3*d^2*c^2*pol
ylog(2,exp(I*(b*x+a)))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^4*cos(b*x+a)*cot(b*x+a)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \cos \left (a+b\,x\right )\,{\mathrm {cot}\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^4 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*x)*cot(a + b*x)^2*(c + d*x)^4,x)

[Out]

int(cos(a + b*x)*cot(a + b*x)^2*(c + d*x)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{4} \cos {\left (a + b x \right )} \cot ^{2}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**4*cos(b*x+a)*cot(b*x+a)**2,x)

[Out]

Integral((c + d*x)**4*cos(a + b*x)*cot(a + b*x)**2, x)

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